Method and system for generating an optimized solution for a process

ABSTRACT

A method, system and computer program product for generating an optimized solution for a process in a manufacturing under a set of constraints. A solution is evaluated under the set of constraints and first violation metrics and first states corresponding to one or more solution elements are generated, based on at least one constraint. At least one candidate solution is generated from the solution. Thereafter, the candidate solution is evaluated at one or more change-points and the solution elements positioned subsequent to the change points. Subsequently, the candidate solution is accepted based on the acceptance criterion. The selected candidate solution is then evaluated again a pre-defined number of times to generate the optimized solution.

REFERENCE TO RELATED APPLICATIONS

This application claims priority of U.S. patent application Ser. No.10/975,751 filed Oct. 28, 2004 and titled “Method and System for Solvingan Optimization Problem” which is incorporated by reference herein.

The application claims priority of U.S. patent application Ser. No.10/975,877 filed Oct. 28, 2004 and titled “Method and system forsequencing and scheduling” which is incorporated by reference herein.

This application claims priority of U.S. Provisional patent applicationSer. No. 61/007,956 filed Dec. 17, 2007 and titled “Method and Systemfor Solving Optimization Problems Using Filters” which is incorporatedby reference herein.

BACKGROUND

This invention relates to generating an optimized solution for a processunder a set of constraints.

Processes are found in all industries. Various examples of the processesin manufacturing organizations include but are not limited to planning,scheduling, and the manufacture of products on an assembly line. Theseprocesses have to be optimized to attain business advantage in terms ofcost and efficiency in execution. Further, these processes are usuallycomplex, and depend on a large number of factors such as variablecapacities of equipment, multiple stages of manufacture, production ofseveral kinds of products using a single resource, manpower or processlimitations, and other factors specific to the manufacturing unit. Inother words, there can be a large number of rules/constraints thatgovern the execution of such processes. A violation of any rule orconstraint directly translates into increased operational costs.Therefore, it is important to optimize such processes by adhering to allconstraints of the environment.

Conventional techniques for generating the optimized solution for theprocess include manual and automated optimization methods. Manualoptimization methods require human intervention. Further, the degree ofoptimization obtained in manual methods largely depends on the skillsand experience of the person involved, as well as the complexity of theprocess involved. Furthermore, manual techniques are alsotime-consuming, cumbersome, and error-prone and result in poor solutionquality. Further, manual techniques fail when the process to beoptimized increases in size and has a large number of constraints.

Automated techniques utilize a defined algorithm to solve theoptimization problem associated with a process. A general class ofautomated methods for optimization is based on the concept of LocalSearch (LS) heuristics. This technique covers a range of combinatorialheuristics, such as simulated annealing, genetic algorithm, taboo searchand evolutionary algorithms.

One such heuristic is provided in the reference titled “Facts,Conjectures and Improvements for Simulated Annealing”, authored by P.Salamon, P. Sibani and R. Frost and published by SIAM Monograph in theyear 2002. Another such heuristic is provided in the reference titled“Genetic Algorithms and Engineering Design”, authored by M. Gen and R.Chang, and published by John Wiley & Sons in the year 1997. Yet anotherheuristic is provided in the reference titled “Modern HeuristicTechniques for Combinatorial Problems”, authored by Colin Reeves andpublished by Halsted Press in the year 1993.

Typically, the LS technique generates the optimized solution by makinglocal moves in the space of solutions. In all LS heuristics, an initialset of one or more solutions is generated by a convenient mechanism. Anyconvenient mechanism can be used because the initial set of solutionsneed not satisfy any constraints; they can therefore even be generatedrandomly. LS heuristics systematically refine the set of solutions bymaking small or local changes. This results in a new set of candidatesolutions. The technique for generating these local changes variesaccording to the LS heuristic used. The set of candidate solutions isthen evaluated to determine violations of the existing constraints.Based on an acceptance criterion that varies according to the heuristicused, one or more of the candidate solutions are accepted. This processis repeated until the candidate solutions do not improve appreciably, ora computational budget is exceeded. In practice, obtaining ahigh-quality solution requires a large number of moves, and thepractical utility of the LS heuristic rests on the efficient evaluationof a large number of candidate solutions (of the order of 10⁶ to 10⁸)against numerous constraints.

Conventionally, candidate solutions are evaluated in their entirety.This requires high-speed computers with large storage capacities, andthe optimization may not run to completion in the available time. If theevaluation is carried out incrementally (i.e., only in the neighborhoodsin which the solution has changed), the method in the current state ofthe art is specific to a particular constraint and depends on the mannerin which the local moves are generated. Thus, if there are K constrainttypes and L schemes for move generation, there is a need to develop K*Lalgorithms, along with commensurate software design, development,testing and maintenance effort. Thus, this approach requires extensivesoftware development which is not cost effective and also timeconsuming. Therefore, it is evident that prolonged computational timesand high expenses hinder/limit the practical use of LS heuristics.

Further, in many manufacturing processes it is desired that certainproduction rules/constraints are applied over certain time intervals,and not over others. Similarly, certain constraints are applicable onlyto a particular model of an automobile or to a particular assembly line.Currently, the optimization techniques, both manual and automated, failto optimize a process by selective application of the constraints.

Accordingly, there is a need for a method that speeds up thecomputational process, along with providing efficient solutions to theprocesses without any manual intervention. Such method should becost-effective, error-free, and should be easily implemented onworkstations and PCs. Further, the method should selectively applyconstraints to generate the optimized solution.

SUMMARY

An object of the present invention is to provide a method for generatingan optimized solution for a process in a manufacturing unit under a setof constraints.

Another object of the invention is to provide a method for selectiveapplication of one or more constraints for generation of an optimizedsolution.

Yet another object of the invention is to provide a method, wherein theincremental evaluation of one or more solutions is essentiallyindependent of the nature of the constraints.

Yet another object of the invention is to provide a method, wherein theevaluation of the solutions is independent of the manner in whichcandidate solutions are generated.

To achieve the objectives mentioned above, the invention provides amethod, system and computer program product for generating an optimizedsolution for a process in a manufacturing unit. The optimized solutionis generated under a set of constraints. For simplicity, first considera LS heuristic in which there is only one candidate solution at eachstep of the computation, for e.g., simulated annealing. The optimizedsolution is generated from a solution representing a sequence of one ormore solution elements. The solution is evaluated, generating firststates and first violation metrics corresponding to each solutionelement in the solution. Thereafter, a candidate solution is generatedby modifying one or more sequence positions in the sequence.Subsequently, the candidate solution is evaluated at one or more changepoints. The change points represent the sequence positions where thesolution elements have been modified. Further, the evaluation isproceeds for one or more solution elements positioned subsequent to thechange points. The evaluation leads to generation of new state and newviolation metric corresponding to each solution element that isevaluated.

The candidate solution is then selected based on an acceptancecriterion. Thereafter, a candidate solution is again generated from theselected candidate solution and is tested again for the acceptancecriterion. The candidate solution is accepted after a pre-defined numberof iterations. The selected candidate solution after a pre-definednumber of iterations is the optimized solution for the process. Further,the method also selectively applies the constraints for the generationof the optimized solution.

This description can be readily extended to LS heuristics in which thereare multiple candidate solutions for e.g., Genetic Algorithms (GA) inwhich at any given computational step there are a population ofcandidate solutions. New solutions are derived from combinations of theexisting set, and the resulting set of solutions can be incrementallyevaluated one at a time as described above.

The method described above have several advantages. The method enablesthe incremental evaluation by integrating the concept of states with theconstraint. This enables the incremental evaluation to be independent ofthe nature of constraints and the manner in which the local changes weremade to a solution/candidate solution. This reduces the softwaredevelopment effort for the scenario described earlier of K constraintsand L schemes for move generation by a factor of K*L. Thus, thissignificantly reduces the cost of developing the software. Further, suchincremental evaluation speeds up the LS heuristic computation, andenables the optimization of processes within the available time. This isan automated optimization that can run without any human intervention.Furthermore, the optimized solution generated is error-free. The methodalso enables selective application of the constraints, which isnecessary to support the full range of real-world constraints.

BRIEF DESCRIPTION OF THE DRAWINGS

The preferred embodiments of the invention will hereinafter be describedin conjunction with the appended drawings provided to illustrate and notto limit the invention, wherein like designations denote like elements,and in which:

FIGS. 1 a and 1 b depict a flowchart illustrating a method forgenerating an optimized solution for a process in a manufacturing unit,in accordance with an embodiment of the invention;

FIG. 2 depicts an example illustrating the method of evaluation of oneor more solution elements for a constraint in accordance with anexemplary embodiment of the invention;

FIGS. 3 a and 3 b depict a flowchart illustrating evaluation of one ormore solution elements of a solution for a constraint in accordance withanother embodiment of the invention;

FIG. 4 depicts an example of evaluating one or more solution elements ofa solution for a constraint in accordance with an exemplary embodimentof the invention;

FIGS. 5 a and 5 b depict a flowchart illustrating evaluation of one ormore solution elements of a solution for a constraint in accordance withanother embodiment of the invention;

FIG. 6 depicts an example of evaluating one or more solution elements ofa solution for a constraint in accordance with an exemplary embodimentof the invention; and

FIG. 7 depicts a flowchart illustrating a method for generating a costassociated with a solution/candidate solution in accordance with anembodiment of the invention.

DETAILED DESCRIPTION OF THE DRAWINGS

For purposes of clarity, the following terms used herein are definedbelow:

Process: A process in a manufacturing unit can be defined as a sequenceof events or tasks to be performed. Various examples of the process canbe assembly line scheduling, distribution planning, vehicle/automatonrouting, and the like. In the context of a manufacturing process, asequence of “jobs” is often the term used.Job: A job is a unit of production. From a scheduling perspective, a jobis an object, an activity, or a process that needs to be scheduled. Forexample, in the process of ‘assembly line scheduling/sequencing formanufacturing 500 cars’ each job can be an automobile.Solution: A solution is a mathematical representation of the process. Inother words, the solution for a process, such as the assembly linescheduling is typically represented as a production schedule. Further,the solution represents sequence of one or more solution elements. Forexample, a solution for scheduling 500 cars on an assembly line isrepresented as {i1, i2, i3, i4 . . . i500}, where each iN (solutionelement) represents an isolated event or job in the sequence. A Ganttchart representing assembly line scheduling is an example of such asolution. Those skilled in the art will appreciate the use of Ganttcharts to depict assembly line production schedules.Solution Element: A solution element represents a single job/event in asolution (sequence of the solution elements). For example, the solutioncan be represented as {i1, i2, i3, . . . , i500}, wherein each ‘iN’represents a solution element. Further, the solution is indexed in thesequence by sequence positions. For example, at first sequence position,i1 is the first solution element.Candidate solution: A candidate solution is an alternate or modifiedsequence derived from the solution. For example, if the solution isrepresented as {i1, i2, i3, i4}, then one candidate solution may be {i1,i4, i3, i2}.Constraint: A constraint is a user-defined rule that the solution shouldadhere to. For example, assembly line scheduling may have a constraintsuch as ‘No more than twenty cars with leather interior are allowed tobe manufactured in a day.’Objective function: An objective function is a mathematical functionwhich is optimized under a set of constraints to attain an optimizedsolution.

$\sum\limits_{i}{w_{i}\left( {\sum\limits_{j}v_{ij}^{s_{i}}} \right)}^{\frac{1}{s_{i}}}$

where, w_(i)=weight assigned to constraint i

-   -   v_(ij)=a violation metric, or a measure of violation j of        constraint i    -   s_(i)=a severity factor or non-negative exponent assigned to        constraint i        State: A state is defined at each element of the solution for a        given constraint. The state at an element encodes the impact of        all other elements on the element for the constraint and        represents the status of the solution element with respect to        the constraint. For example, if the constraint were a limit on        the number of cars manufactured with leather interior in a given        day, the state of this constraint for the nth solution element        will be the number of cars manufactured with leather interior        prior to, for that day.        Attribute: An attribute represents an associated property with        the solution element. For example, a car may have associated        attributes such as model, variant, color, leather interiors, and        the like.

It will be appreciated by those skilled in the art that although thedefinitions have been provided in the context of manufacturing a set ofcars, the embodiments of the invention can be implemented across a widevariety of different processes, both within the context of massmanufacturer and beyond.

Turning now to FIGS. 1 a and 1 b, illustrates a flowchart of a methodfor generating an optimized solution for a process in a manufacturingunit, in accordance with embodiments of the invention.

In various embodiments of the invention, an optimized solution isachieved by iteratively optimizing various candidate solutions derivedfrom an initial solution (or initial set of solutions). The initialsolution can simply be a random sequence of solution elements that canbe expected to violate a maximum number of constraints. Alternately,simple heuristics can be used to generate the initial solution, forexample, those that attempt to satisfy one or two high priorityconstraints. The main requirement for generating the initial solution isthat it is requires minimal computational time. By deriving one or morecandidate solutions and iteratively refining the candidate solutions fora pre-defined number of times, a near optimal solution may be achieved.

At 102, an initial solution, herein after referred to as the solution,is evaluated under a set of constraints. The set of constraints aredecided based on the sequence that is being optimized. For example, if asolution for an assembly line is being generated, then the constraintsconsidered are the constraints that effect the scheduling/sequencing ofthe assembly line. Further, as described above, the solution includes asequence of one or more solution elements. In an embodiment of theinvention, the evaluation is performed to check whether each solutionelement in the sequence adheres to the set of constraints.

In keeping with the example, assume the process is the sequencing of aset of cars (i.e., 500 cars) to be manufactured. Each car may be builtwith different features (i.e., moon roof, body color, type of interior).A set of constraints may be placed on this process. The constraints maybe caused by a wide variety of reasons. For example, a constraint may be“the number of cars manufactured with leather interior should not exceedtwenty per day”. This constraint may exist because the facility at whichthe manufacturing is being conducted may only have enough dailyinventories of leather interior packages to support twenty or so cars.

Another constraint may be that ‘a white color’ cannot follow carspainted dark colors. This constraint may be in place because thepainting process may contaminate a white color when the applicatorsswitch from dark paint to white paint.

Yet another constraint may be ‘back to back moon roofs are not allowed’or ‘export cars should be built before Wednesday’. As one can see, thereare a seemingly infinite number of constraints that may be defined andimplemented. Collectively these constraints can be defined as the set ofconstraints with which the solutions must be evaluated.

Returning to Step 102, the initial solution is evaluated to checkwhether it adheres to the set of constraints. That is, whether the arrayof solution elements and their individual attributes (i.e., the 3^(rd)car may come with a moon roof and painted white) adhere to the set ofconstraints. In various embodiments of the invention, the initialsolution is generated by using a pre-defined algorithm. For example, theinitial solution may be generated by using a random generation method,or a well-known greedy heuristic that considers only a small number ofhigh priority constraints. It should be appreciated by those skilled inthe art that any number of algorithms could be used to create theinitial solution and generally any particular algorithm is beyond thescope of the present invention.

At 104, a first violation metric and a first state are generated foreach of the solution elements in the solution. In other words, eachsolution element at the corresponding each sequence position isevaluated for each constraint of the set of constraints. It may beapparent to any person skilled in the art that the solution isrepresented by a sequence.

A violation metric is a measure of the violation of a constraint to aparticular extent. In various embodiments of the invention, theviolation metric is a numeric value. In other embodiments, the violationmetric may be Boolean (i.e., given a value of ‘1’ when a constraint isviolated and a value of ‘0’ when the constraint is not violated.

A state of the solution element as described above represents the statusof the solution element. Further, each state for the solution element iscalculated based on the state of the solution element positioned priorto the solution element. For example, if a solution element at the35^(th) sequence position in the solution has to be checked whether itis adhering to a constraint, such as, ‘cars having leather interiorsshould not exceed 20 per day’, then the state at the 34^(th) sequenceposition in a sequence has to be considered to check the adherence.Therefore, if there are already 20 ‘leather interior cars’ positioned inthe initial 34 cars in the sequence, then the state of solution elementat the 34^(th) sequence position will be ‘20 leather interior cars’ andbased on the car at the 35^(th) position in the sequence state isupdated. Thus, if solution element at 35^(th) sequence position is alsoa ‘leather interior car’, then the state of the solution element at35^(th) sequence position will be ‘21 leather interior cars’ and if thesolution element at 35^(th) sequence position car is not a leatherinterior cars then state of the solution element at 35^(th) sequenceposition for the constraint remains the same as the state of thesolution element at the 34^(th) sequence position, i.e. ‘20 leatherinterior cars.’

It may be apparent that the first violation metric and the first statefor each constraint at each sequence position is generated sequentially,i.e. one after the other. In various embodiments of the invention, theevaluation is started with a null state for each constraint and then itis replaced with the first state based on the evaluation.

At 106, a candidate solution is generated based on the solution. Thecandidate solution is generated by modifying the sequence at least onesequence position.

In an embodiment of the invention, the solution elements atcorresponding sequence positions can be randomly swapped in the solutionto generate the candidate solution. In various embodiments of theinvention, the candidate solution is generated by using mathematicalfunctions, also termed as operators. For example, an operator can be ‘aswap function’ which generates two random numbers indicating thesequence positions that have to be swapped in the sequence. Similarly,there may be ‘an insert function’, ‘a shift function’, and ‘a deletefunction’. It may be apparent to any person skilled in the art thatthere may be various other ways to generate the candidate solution, inother words, to generate a new sequence.

For example, in a sequence of 500 solution elements, a solution elementat the 300^(th) sequence position can be swapped with a solution elementat the 200^(th) sequence position to generate the candidate solution.Further, the candidate solution is a characteristic of SimulatedAnnealing. It may be apparent to any person skilled in the art that thecandidate solution that is generated based on the solution has to begenerated by the same pre-defined algorithm which is used for generatingthe solution.

At 108, the candidate solution is evaluated. The candidate solution isevaluated at a. each solution element corresponding to each change pointand b. one or more solution elements positioned subsequent to eachchange point.

A change point can be defined as a sequence position at which thesolution element in the solution is modified, i.e. replaced, swapped,added, deleted, and the like. Following the above example, the changepoints will be the sequence position 200 and sequence position 300.Thus, the candidate solution is evaluated first at sequence position200. A solution element at the sequence position 200 is evaluated basedon the first state at the sequence position 199. For example, atsequence position 199, the first state corresponding to the constraint‘cars having leather interiors should not exceed 20 per day’ is 20, i.e.199 initial set of cars to be manufactured already includes 20 cars withleather interiors. Further, a first state corresponding to theconstraint ‘dark colors cannot follow light colors’ is ‘Dark’, i.e. atsequence position 199 there is already a ‘dark car’.

Further, if the solution element, i.e. swapped car/job, now at thesequence position 200 has attributes as ‘Leather Interiors’ and ‘WhiteColored’, then a new state corresponding to the constraint ‘cars havingleather interiors should not exceed 20 per day’ will be now 21 and a newstate corresponding to the constraint ‘dark colors cannot follow lightcolors’ will be now ‘Light’. However both the constraints have beenviolated, hence a new violation metric associated with each constraintwill be generated simultaneously. It may be apparent to any personskilled in the art that similarly, a new state and a new violationmetric can be generated for any number of constraints.

Thereafter, since the sequence position 200 is altered, it may beapparent to any person skilled in the art that one or more solutionelements positioned subsequent to the change point will also have animpact of the new state at sequence position 200. Therefore, anevaluation of the subsequent solution elements (i.e., sequence position201) is conducted to determine its new state(s) and any new violationmetrics as a result of the altered state(s).

The evaluation is stopped when the state(s) in a given sequence position(e.g., perhaps sequence position 215) does not change.

As is known by those skilled in that art, considering classical statemachine theory, the evaluation is completely determined by (i) thesolution element and (ii) the state at that point. Thus, incrementalevaluation proceeds as long as the candidate solution is logicallyunequal with the solution, or the state and the new state are unequal.For example, if at sequence position 215 a solution element has anoriginal state as ‘Dark’, and a new state (based on the attributes ofthe solution element at sequence position 215 and the state of thesolution element at sequence position 214) is ‘Dark’, then the state ofthe solution element at sequence position 216 will be the same as itwere before. Therefore, there is no need to continue evaluation pastsequence position 216, as the state of all subsequent sequence positionswould be the same as their original states.

As one can see, based upon features of the present invention, ifsequence position 200 is altered as the result of generating a candidatesolution, computation on the previous 199 elements is not necessary.Furthermore, computation on the subsequent elements may or may not benecessary (or at least limited); depending on the impact the change tosequence position 200 brings to the remainder of the sequence. Oneskilled in the art will appreciate how this approach provides for moreefficiency via reduced computation compared to the conventionaloptimization techniques.

To complete the example above, evaluation of the change in state(s) ofthe solution element at sequence position 300 (now swapped with thesolution element at the sequence position 200) is necessary in much thesame manner as the evaluation of the solution element at sequenceposition 200. In various embodiments such an evaluation is termed as‘Incremental Evaluation’.

At 110, the candidate solution is tested for an acceptance criterion. Inan embodiment of the invention, the acceptance criterion is selectingthe candidate solution when a cost associated with the candidatesolution is less than a pre-defined cost. In various embodiments of theinvention, the violation metric associated with each solution element isconsidered for calculating the cost. It may be apparent to any personskilled in the art that the cost is generated from optimizing anobjective function associated with the process. Calculation of the costfrom the violation metrics has been explained in detail in conjunctionwith FIG. 7.

At 110, if the calculated cost is not less than the pre-defined cost(i.e., the acceptance criterion is not met), then the candidate solutionis not selected and a new candidate solution is generated at step 106.If the acceptance criterion is met, then the candidate solution isselected. Subsequently, at 112, the new violation metric and the newstate corresponding to each constraint for each solution elementassociated with the selected candidate solution are updated. In otherwords, the first states and the first violation metrics are replaced bythe corresponding new states and the new violation metrics.

In another embodiment of the invention, the candidate solution isselected probabilistically when the candidate solution has the costgreater than the pre-defined cost. To accept the candidate solutionprobabilistically a random number is generated. Thereafter, it ischecked whether the random number generated is less than e^(−d/T) and ifso, the candidate solution is selected.

The method proceeds by determining whether the stopping criterion is met(Step 114). In one embodiment of the invention, the stopping criterionstops the evaluation after a pre-defined number of iterations. Inanother embodiment of the invention, the stopping criterion stops theevaluation procedure when there is no further improvement in the costfunction. The amount of improvement may be a variable function, such asx % improvement over n iterations. Failure to accomplish this iterativeimprovement would result in the stopping criterion being met.

At 114, if it is determined that the stopping criterion is not met then,at 116, the solution is replaced with the candidate solution selected at110 and the process of evaluation is carried again from step 106, i.e. acandidate solution is again generated based on the selected candidatesolution. These steps, i.e. 106, 108, 110, 112 and 114 are repeateduntil a stopping criterion is met.

Once the stopping criterion is met, the selected candidate solution isthe optimized solution for the process.

In another embodiment of the invention, the LS heuristic used is aGenetic Algorithm. It may be apparent to any person skilled in the artthat, Genetic Algorithms work with multiple candidate solutions at anygiven computational stage. In an exemplary embodiment of the invention,say M=100 number of solutions are generated. Thereafter, steps 102, 104,106, 108, 110, 112, 114 and 116 are performed for each solution. Newcandidate solutions are generated from the existing population usingwell-known operators such as Crossover or Mutation. Once, a new set ofcandidate solutions are generated, the incremental evaluation asdescribed above can be applied to each of the candidates individually.Subsequently, top 100 candidate solutions are retained.

In other embodiments of the invention, the LS heuristic used is taboosearch, and evolutionary algorithms. It may be apparent to any personskilled in the art that Simulated Annealing, Genetic Algorithms, taboosearch and evolutionary algorithms are examples of Local Search (LS)Heuristics. In various embodiments of the invention, the solutionstarting from an initial solution is systematically refined using one ofthe above mentioned algorithms.

FIG. 2 depicts an example illustrating the method of evaluation of oneor more solution elements for a constraint in accordance with anexemplary embodiment of the invention.

An initial solution comprising a sequence of elements: ‘A B C D E F’.The sequence represents a series of cars to be manufactured. An optimumsequence or schedule of manufacture is to be determined, provided theuser-defined constraint that back to back moon roofs are not permitted.In this example, cars ‘B’, ‘D’, and ‘E’ require moon roofs and ‘A’, ‘C’,and ‘F’ do not. A state corresponding to the moon roof car isrepresented by ‘MR’ and a state corresponding to the non-moon roof caris represented by ‘NMR’. Following the flowchart described in FIG. 1, at102 the solution is evaluated for the constraint. Further, as explainedthe initial state corresponding to sequence position 1 is a ‘null state’(not shown in FIG. 2). In other words, given that there are no solutionelements prior to the first element, there is no state information thathas to be considered for the very first element. Thereafter, a firststate and a first violation metric are generated for each solutionelement in the sequence.

Evaluation starts from the first solution element (A) at the firstsequence position. Since, A is a non moon roof car; the state is updatedwith ‘NMR’. Further, the constraint is not violated; hence the violationmetric is ‘Zero/0’. Thereafter, B is a moon roof car and thereforecorresponding first state is updated as ‘MR’. Similarly, C and D arealso evaluated. However, E is a moon roof car, and the state consideredfor evaluation of E is the first state of D, i.e. ‘MR’, therefore theconstraint at E is violated, hence, a violation metric (X) is generatedand the first state is updated as ‘MR’. Similarly F is also evaluatedand the corresponding first state and the first violation metric areupdated. Table 1 represents the corresponding first states and the firstviolation metrics corresponding to the solution elements of thesolution.

Further, as described in FIG. 1, a candidate solution is generated atstep 106. Following the current example, solution element at the secondsequence position is swapped with the solution element at the sixthsequence position. Thus, the candidate solution is now represented as ‘AF C D E B’, F and B being the solution elements at the change points.

F is re-evaluated based on the state of A, and since, the F is non moonroof car, and the first state is replaced with a new state (NMR).Thereafter, C is evaluated based on the state of F. Since, C is anon-moon roof car the new state generated based on the new state of F is‘NMR’. It may be apparent that the first state of C and new state of Cremains the same (shown in FIG. 2); hence the evaluation thereafter willprovide the same state and same violation metric. Thus, D and E are notevaluated. The evaluation starts at the next change point, i.e. B andthe corresponding new state and the new violation metric are accordinglyupdated. Table 2 details the states and the violation metrics associatedwith the candidate solution. This process would continue until astopping criterion was met (i.e., a sequence that did not violate theconstraint).

If we can consider a computation event to be a determination of aviolation metric, then this iteration required three computation events(determining the violation metric for the 2^(nd), 3^(rd), and 6^(th)event in the solution. Were a prior art approach used, computation wouldbe required to determine the 2^(nd), 3^(rd), 4^(th), 5^(th), and 6^(th)event in the solution. One can appreciate the efficiency in computationover a number of iterations.

The method described in the FIG. 1 evaluates each solution elementirrespective of the attributes associated with the solution element atthe change points and the subsequent solution elements positioned afterthe change point in the sequence. However, there might be a possibilityto apply constraints selectively based on the attributes associated withthe solution element. Such a method of selective application ofconstraints is explained in detail in conjunction with FIG. 3, FIG. 4,FIG. 5 and FIG. 6.

FIGS. 3 a and 3 b depict a flowchart illustrating the evaluation of oneor more solution elements of a solution for a constraint in accordancewith another embodiment of the invention.

In various embodiments of the invention, a constraint is selectivelyapplied to a solution element. For example, in a manufacturing unit, aspecific constraint is applied to only a specific model of car which isbut one of a handful of models to be manufactured in one process. Suchselective application of the constraint helps in reduction ofunwanted/undesired evaluation of the solution element and thus improvesthe speed of the evaluation of the solution leading to faster generationof the optimized solution. The method of selectively applying theconstraint on the solution element is explained in detail below.

At 302, at least one attribute of the solution element of the solutionis matched with a pre-defined attribute. If at 302, it is determinedthat the at least one attribute of the solution element does not matchwith the pre-defined attribute, then at 304 the solution element is notevaluated and a corresponding state of the solution element is updated.

Further, if at 302 it is determined that the attribute of the solutionelement matches the pre-defined attribute, then at 306, the solutionelement is evaluated for the constraint. Evaluation of the solutionelement is explained in detail in conjunction with FIG. 1.

At 308, it is determined if there is a violation of the constraint. Ifat 308, it is determined that there is no violation, then at 310, aviolation metric is not generated and a corresponding state of thesolution element is updated. Further, if at 208 it is determined thatthere is a violation of the constraint then at 312 a violation metric isgenerated and the state is updated accordingly.

Thus, the constraint is selectively applied to the solution elementbased on the attribute associated with it. Further, based on theviolation metric generated the optimized solution varies accordingly.Therefore, it must be decided whether the violation metric that isgenerated at 310 should be considered for generating the optimizedsolution.

Thus, at 314, it is determined if the generated violation metric has tobe selected. The violation metric is selected based on a firstpre-defined criterion. In an embodiment of the invention, the firstpre-defined criterion may be selecting the violation metric if thegenerated violation metric is in a pre-defined range. In anotherembodiment of the invention, the first pre-defined criterion isselecting the violation metric based on the time corresponding to thesolution element at the sequence position for which the violation metricis generated. For example, in manufacturing units, the first pre-definedcriterion may be to discard the generated violation metric during thenight shifts.

If at 314 the violation metric is selected, then the violation metric istransformed into a cost at 316. Further, if the violation metric is notselected at 314, then at 318 the violation metric is discarded. In otherwords, the generated violation metric is not used to generate the cost.Similarly, each solution element is evaluated. The flowchart describedabove is further explained with an example explained in conjunction withFIG. 4.

FIG. 4. depicts an example of evaluating one or more solution elementsof a solution for a constraint in accordance with an exemplaryembodiment of the invention.

An initial solution for scheduling four cars for manufacturing on asingle assembly line is considered. The solution is represented by asequence ‘A A B B’. Herein, A represents ‘white colored cars’ and Brepresents ‘red colored cars’. The attributes for each solution elementof the sequence are for example, color of the car, model of the car . .. , etc. In this example, all the cars in the sequence have moon roofs.

The solution is evaluated for a constraint, ‘back to back moon roofs arenot allowed’. In addition, the constraint has to be applied to redcolored cars only. Accordingly, the pre-defined attribute is ‘redcolor’. State corresponding to the moon roof car is represented by ‘MR’.

As explained earlier in FIG. 3, the attribute of the first solutionelement of the sequence, i.e., A (white colored car) is checked with thepre-defined attribute. Since, A is a white colored car, the firstpre-defined attribute criterion is not met, and accordingly, firstsolution element (A) in the sequence is not evaluated for theconstraint. However, the corresponding state is updated. It may beapparent to any person skilled in the art that since A is the firstsolution element in the sequence and is not evaluated, the state isupdated as ‘null state/0’. Thereafter, as the second solution element inthe sequence is also a white colored car, the second solution element isnot evaluated. Further, the state of the second solution element isupdated as the state for the first solution element, i.e. null state/0.Furthermore, no violation metric is generated for the both the solutionelements (A and A).

Thereafter, third solution element B is evaluated as it is a red coloredcar and it matches the pre-defined attribute. Further, the state and thecorresponding violation metrics are updated. The state is updated as‘MR’, since the third solution element represents a moon roof car.Furthermore, the violation metric generated for the third solutionelement is ‘zero’ as the constraint is not violated. Subsequently, thefourth solution element B is also evaluated and the corresponding stateand violation metric is updated. The state is updated as ‘MR’. However,the fourth solution element violates the constraint. Thus, a violationmetric ‘X’ is generated accordingly.

FIGS. 5 a and 5 b is a flowchart illustrating evaluation of one or moresolution elements of a solution for a constraint in accordance withanother embodiment of the invention.

In an embodiment of the invention, there may be more than one assemblyline in a manufacturing unit. Further, each assembly line manufacturesthe car in a separate fashion by adhering to a separate set ofconstraints. For example, a particular constraint need not be applicableto a first assembly line but is applicable to a second assembly line inthe manufacturing unit. Further, an evaluation of a solution elementunder such environment is explained below.

At 502, a first attribute of a solution element is matched with a firstpre-defined attribute. For example, the first attribute may be ‘assemblyline 1’. If at 502, it is determined that the first attribute of thesolution element does not match with the first pre-defined attribute,then at 504, the solution element is not evaluated and correspondingstate of the solution element is updated.

Further, if the first attribute of the solution element matches thefirst pre-defined attribute at 502, then at 506, a second attribute ofthe solution element is matched with a second pre-defined attribute ofthe solution element. For example, the second attribute may be ‘color’of the car. If at 506 it is determined that the second attribute of thesolution element does not match the second pre-defined attribute, thenat 508, the solution element is not evaluated and the correspondingstate of the solution element is updated. However, if the secondattribute of the solution element matches the second pre-definedattribute at 506, then the solution element is evaluated for theconstraint at 510. The evaluation of the solution element has beenexplained in detail in conjunction with FIG. 1 and FIG. 2.

At 512, it is determined if the constraint is violated by the solutionelement. If at 512, it is determined that the constraint is notviolated, then at 514 a violation metric is not generated andcorresponding state of the solution element is updated. However, if at512, it is determined that the constraint is violated by the solutionelement, then at 516 a violation metric is generated and a correspondingstate of the solution element is updated.

Thus, at 518, it is determined if the generated violation metric has tobe selected. The violation metric is selected based on a firstpre-defined criterion. In an embodiment of the invention, the firstpre-defined criterion may be selecting the violation metric when thegenerated violation metric is in a pre-defined range. In anotherembodiment of the invention, the first pre-defined criterion isselecting the violation metric based on the time the violation metric isgenerated. For example, in manufacturing units, the first pre-definedcriterion may be to discard the generated violation metric during thenight shifts.

If at 518 the violation metric is selected, then the violation metric istransformed into a cost at 520. Further, if the violation metric is notselected at 518, then at 522 the violation metric is discarded. In otherwords, the generated violation metric is not used to generate the cost.Similarly, each solution element is evaluated. The flowchart describedabove is further explained with an example explained in conjunction withFIG. 6.

FIG. 6 depicts an example of evaluating one or more solution elements ofa solution for a constraint in accordance with an exemplary embodimentof the invention.

An initial solution for scheduling six cars for manufacturing on twodifferent manufacturing assembly lines is considered. The solution isrepresented by a sequence ‘A1 A2 B1 B2 A3 A4’. Herein, A1, A2, A3 and A4represent cars associated with assembly line 1 (Line1) and, B1 and B2represent cars associated with assembly line 2 (Line2). Further, A1 andA4 are ‘white colored cars’, and A2, B1, B2, A3 are ‘red colored cars’.Therefore, the attributes for each solution element of the sequence arefor example, color of the car, associated assembly line . . . , etc.Furthermore, A4 is a non moon roof car, and A1, A2, B1, B2 and A3 aremoon roof cars.

As explained earlier, the solution is evaluated for a constraint. Theconstraint considered in the current example ‘Back to back moon roofsare not allowed’ is applicable only to red cars on ‘Line1’. Accordingly,the first pre-defined attribute is ‘Line 1’ and the second pre-definedattribute is ‘red.’ State corresponding to the moon roof car isrepresented by ‘MR’.

Following the current example and as described in FIG. 5, the firstattribute of first solution element A1 matches the first pre-definedattribute (Line1) however, the A1 is a white colored car, hence it isnot evaluated and the state is updated as ‘0’ and violation metric isupdated as ‘Zero/O’. It may be apparent since A1 is the first solutionelement, its initial state must be a ‘null state/O’ and since it is notevaluated, the state is updated as ‘null state/0’ again. Further, theviolation metric is zero as A1 was not evaluated for the constraint.

Thereafter, A2 matches the first pre-defined attribute and since A2 is a‘red colored car’, it is evaluated and corresponding state and violationmetric is updated. Further, since A2 is a moon roof car, the state isupdated as ‘MR’. However, no violation metric is generated as theconstraint is not violated.

After the evaluation of A2, B1 and B2 are not evaluated as B1 and B2 donot belong to ‘Line1’, thereby not satisfying the first pre-definedattribute. Further, states corresponding to the solution elements areupdated. In various embodiments of the invention, the state to beconsidered for the evaluation of the solution element is reset for thesolution element when solution element to be evaluated is from adifferent assembly line. It may be apparent to any person skilled in theart that the state to be considered for evaluation of the solutionelement will differ for two different assembly lines. In the currentexample, since B1 is the first solution element in the sequenceassociated with the assembly line 2, its initial state is considered as‘null state/0’. However, B1 is not evaluated and the state is updated as‘null state/0’ again. Similarly, state of B2 is updated based on B1.

Further, the state considered for evaluation of the solution elementassociated with an assembly line is considered as the state of theprevious solution element of the same assembly line. Thus, when the nextsolution element A3 matches the first pre-defined attribute, the stateis reset and the state considered for evaluation of A3 is ‘MR’ (state ofA2). Further, A3 is a ‘red colored car’ and hence it is evaluated and acorresponding state and violation metric is generated. Since, theconstraint is violated a violation metric X (depicted in FIG. 6), isgenerated and the state is updated as ‘MR’. Similarly, A4 is notevaluated since the second attribute of A4 does not match with thesecond pre-defined attribute.

FIG. 7 depicts a flowchart illustrating a method for generating a costassociated with a solution/candidate solution, in accordance with anembodiment of the invention.

In various embodiments of the invention, a cost is generated for asolution/candidate solution. Further, the cost is generated based on aviolation metric associated with each solution element for a constraint.For example, the violation metric of the solution element is ‘20’.Further, the constraint may have a pre-defined range, such as min=5;max=10. The generation of the cost is explained in detail below.

At 702, each violation metric associated with each solution element ofthe solution is transformed based on a pre-defined function. Theviolation metric is transformed to ‘10’ based on a pre-defined function.In the current example, the violation metric deviates by ‘10’ (20-10) asmax=10 and violation metric=20. A sample pre-defined function is to takethe square of the deviation, 10̂2=100. More generally, the pre-definedfunction f(x) can be of the form x̂n, where n is the severity factor=0,1, 2 and so on. In various embodiments of the invention, the violationmetric is numeric.

At 704, each transformed violation metric is normalized. In variousembodiments of the invention, the transformed violation metrics arenormalized in order to remove the effect of intrinsic variations in themagnitude of the violation metric. In an exemplary embodiment of theinvention, the violation metric is transformed by raising it to thepower of the severity factor and then divided by the total penalty ofthe constraint. One example of the total penalty that can be used is asfollows:

${{Total}\mspace{14mu} {penalty}} = {\sum\limits_{i}^{K}{w_{i}\left( {\sum\limits_{j}^{m_{i}}d_{j}^{n_{i}}} \right)}^{1/n_{i}}}$

Where K=number of constraints

-   -   w_(l)=weight or scaling factor for constraint i    -   m_(i)=number of violations for constraint i    -   d_(j)=jth deviation for constraint i    -   n_(i)=severity factor for constraint i

Further, the severity factor is pre-defined by a user. In variousembodiments of the invention, the transformed violation metric isnumeric

At 706, each normalized violation metric is scaled based on acorresponding priority associated with the constraint. In the currentexample, the normalized violation metric is scaled based on apre-defined formula, i.e. the result from 704 * priority associated withthe constraint. In various embodiments of the invention, the result isnumeric. In an embodiment of the invention, the priority of the solutionelement is numeric value pre-defined by the user.

At 708, summing each scaled violation metric generated for each solutionelement corresponding to each constraint generates the cost associatedwith the solution/candidate solution.

In an embodiment of the invention, the method described in FIG. 1, FIG.3, FIG. 5 and FIG. 7 is performed by a processor of a computer system.Further, the states and the violation metrics are stored in a memory ofthe computer system. Furthermore, the methods described in FIG. 1, FIG.3, FIG. 5 and FIG. 7 are implemented in JAVA. It may be apparent to anyperson skilled in the art that, the methods can be implemented inprogramming languages like C, C++, C#, PERL, and the like.

In another embodiment of the invention, the method described in FIG. 1,FIG. 3, FIG. 5 and FIG. 7 can also be implemented in hardware. It may beimplemented in an integrated circuit or part of an integrated circuit,such as an application specific integrated circuit, field programmablegate arrays, programmable logic arrays, and full-custom integratedcircuits.

The method described above have several advantages. The method enablesthe incremental evaluation by integrating the concept of states with theconstraint. This enables the incremental evaluation to be independent ofthe nature of constraints and the manner in which the local changes weremade to a solution/candidate solution. Thus, this reduces the cost ofdeveloping the software and thereby it is less time consuming. Further,such incremental evaluation speeds up the computational speed and doesnot require any human intervention. Furthermore, the optimized solutiongenerated is error-free. The method also enables selective applicationof the constraints, which avoids unnecessary evaluation and thusincreasing the computational speed.

The methods and systems as described above can be implemented insoftware (e.g., firmware), hardware, or a combination thereof. In thecurrently contemplated best mode, the system is implemented in software,as an executable program, and is executed by a special or generalpurpose digital computer, such as a personal computer (PC;IBM-compatible, Apple-compatible, or otherwise), workstation,minicomputer, or mainframe computer.

Generally, in terms of hardware architecture, the computer includes aprocessor, memory, and one or more input and/or output (I/O) devices (orperipherals) that are communicatively coupled via a local interface. Thelocal interface can be, for example but not limited to, one or morebuses or other wired or wireless connections, as is known in the art.The local interface may have additional elements, which are omitted forsimplicity, such as controllers, buffers (caches), drivers, repeaters,and receivers, to enable communications. Further, the local interfacemay include address, control, and/or data connections to enableappropriate communications among the aforementioned components.

The processor is a hardware device for executing software, particularlythat stored in memory. The processor can be any custom made orcommercially available processor, a central processing unit (CPU), anauxiliary processor among several processors associated with thecomputer, a semiconductor based microprocessor (in the form of amicrochip or chip set), a microprocessor, or generally any device forexecuting software instructions.

The memory can include any one or combination of volatile memoryelements (e.g., random access memory (RAM, such as DRAM, SRAM, SDRAM,etc.)) and nonvolatile memory elements (e.g., ROM, hard drive, tape,CDROM, etc.). Moreover, the memory may incorporate electronic, magnetic,optical, and/or other types of storage media. Note that the memory canhave a distributed architecture, where various components are situatedremote from one another, but can be accessed by the processor.

The software in memory may include one or more separate programs, eachof which comprises an ordered listing of executable instructions forimplementing logical functions. In keeping with the examples provided,the software in memory includes the optimization system in accordancewith the present invention and a suitable operating system. Theoperating system essentially controls the execution of other computerprograms, such as the optimization system, and provides scheduling,input-output control, file and data management, memory management, andcommunication control and related services.

The optimization system is a source program, executable program (objectcode), script, or any other entity comprising a set of instructions tobe performed. When a source program, then the program needs to betranslated via a compiler, assembler, interpreter, or the like, whichmay or may not be included within the memory so as to operate properlyin connection with the O/S. Furthermore, the optimization system can bewritten as (a) an object oriented programming language, which hasclasses of data and methods, or (b) a procedure programming language,which has routines, subroutines, and/or functions, for example but notlimited to, C, C++, Pascal, Basic, Fortran, Cobol, Perl, Java, and Ada.

The I/O devices may include input devices, for example but not limitedto, a keyboard, mouse, scanner, microphone, etc. Furthermore, the I/Odevices may also include output devices, for example but not limited to,a printer, display, etc. Finally, the I/O devices may further includedevices that communicate both inputs and outputs, for instance but notlimited to, a modulator/demodulator (modem; for accessing anotherdevice, system, or network), a radio frequency (RF) or othertransceiver, a telephonic interface, a bridge, a router, etc.

If the computer is a PC, workstation, or the like, the software in thememory may further include a basic input output system (BIOS\. The BIOSis a set of essential software routines that initialize and testhardware at startup, start the O/S, and support the transfer of dataamong the hardware devices. The BIOS is stored in ROM so that the BIOScan be executed when the computer is activated.

When the computer is in operation, the processor is configured toexecute software stored within the memory, to communicate data to andfrom the memory, and to generally control operations of the computerpursuant to the software. The optimization system and the O/S, in wholeor in part, but typically the latter, are read by the processor, perhapsbuffered within the processor, and then executed.

When the optimization system is implemented in software it should benoted that the optimization system can be stored on any computerreadable medium for use by or in connection with any computer relatedsystem or method. In the context of this document, a computer readablemedium is an electronic, magnetic, optical, or other physical device ormeans that can contain or store a computer program for use by or inconnection with a computer related system or method. The optimizationsystem can be embodied in any computer-readable medium for use by or inconnection with an instruction execution system, apparatus, or device,such as a computer-based system, processor-containing system, or othersystem that can fetch the instructions from the instruction executionsystem, apparatus, or device and execute the instructions. In thecontext of this document, a “computer-readable medium” can be any meansthat can store, communicate, propagate, or transport the program for useby or in connection with the instruction execution system, apparatus, ordevice. The computer readable medium can be, for example but not limitedto, an electronic, magnetic, optical, electromagnetic, infrared, orsemiconductor system, apparatus, device, or propagation medium. Morespecific examples (a non-exhaustive list) of the computer-readablemedium would include the following: an electrical connection(electronic) having one or more wires, a portable computer diskette(magnetic), a random access memory (RAM) (electronic), a read-onlymemory (ROM) (electronic), an erasable programmable read-only memory(EPROM, EEPROM, or Flash memory) (electronic), an optical fiber(optical), and a portable compact disc read-only memory (CDROM)(optical). Note that the computer-readable medium could even be paper oranother suitable medium upon which the program is printed, as theprogram can be electronically captured, via for instance opticalscanning of the paper or other medium, then compiled, interpreted orotherwise processed in a suitable manner if necessary, and then storedin a computer memory.

In an alternative embodiment, where the optimization system isimplemented in hardware, the optimization system can implemented withany or a combination of the following technologies, which are each wellknown in the art: a discrete logic circuit(s) having logic gates forimplementing logic functions upon data signals, an application specificintegrated circuit (ASIC) having appropriate combinational logic gates,a programmable gate array(s) (PGA), a field programmable gate array(FPGA), etc.

While the preferred embodiments of the invention have been illustratedand described, it will be clear that the invention is not limited tothese embodiments only. Numerous modifications, changes, variations,substitutions and equivalents will be apparent to those skilled in theart, without departing from the spirit and scope of the invention, asdescribed in the claims.

1. A method for generating an optimized solution for a process in amanufacturing unit, the method comprising: a. evaluating at least onesolution comprising plurality of solution elements associated with theprocess based on a set of constraints, the at least one solutionrepresenting a sequence of the plurality of solution elements, eachsolution element having one or more attributes, the at least onesolution being generated using a pre-defined algorithm, wherein the atleast one solution element is evaluated based on the corresponding atleast one attribute; b. generating a first violation metric and a firststate for each solution element, the first state and the first violationmetric being generated corresponding to at least one constraint based ona first pre-defined criterion, wherein the first state is generatedbased on a first state of a solution element positioned prior to thesolution element in the sequence for the at least one constraint; c.generating a candidate solution by modifying at least one sequencepositions of one or more solution elements in the at least one solution;d. evaluating the candidate solution at: i. each solution elementcorresponding to each change point, a solution element at each changepoint being evaluated for the at least one constraint based on a firststate of a solution element positioned prior to the solution element inthe sequence, and the corresponding at least one attribute, wherein anew state and a new violation metric corresponding to the at least oneconstraint for the solution element at the corresponding each changepoint is generated; ii. one or more solution elements positionedsubsequent to each change point corresponding to the at least oneconstraint, wherein the evaluation at the one or more solution elementsis performed till at least one solution element of the one or moresolution elements has a new state different from the corresponding firststate, wherein each solution element is evaluated based on thecorresponding at least one attribute and a violation metric is generatedfor each of evaluated solution element; e. testing the candidatesolution based on an acceptance criterion; f. updating the firstviolation metric and the first state associated with each solutionelement corresponding to each change point and the at least one solutionelement of the one or more subsequent solution elements when thecandidate solution is selected after testing; g. repeating steps c, d, eand f for a pre-defined number of times, wherein the at least onesolution is replaced with the selected candidate solution; and h.outputting the optimized solution, wherein the optimized solution is theselected candidate solution after the pre-defined number of iterations.2. The method according to claim 1, wherein at least one solutionelement of the at least one solution is evaluated when at least oneattribute of the at least one solution element matches at least onepre-defined attribute, the at least one solution element being evaluatedfor a constraint.
 3. The method according to claim 1, wherein at leastone solution element of the candidate solution is evaluated when atleast one attribute of the at least one solution element matches apre-defined attribute, the at least one solution element being evaluatedfor a constraint.
 4. The method according to claim 1, wherein updatingthe first violation metric is performed based on the first pre-definedcriterion.
 5. The method according to claim 1, wherein the acceptancecriterion comprises a cost associated with the candidate solution beingless than or equal to a pre-defined cost, the calculation of the costcomprises: a. transforming each violation metric associated with eachsolution element of the candidate solution, each violation metric beingtransformed based on a pre-defined function, each violation metric beingassociated with a corresponding constraint; b. normalizing eachtransformed violation metric; c. scaling each normalized violationmetric based on a priority associated with the corresponding constraint;and d. summing the scaled violation metric corresponding to eachconstraint to generate the cost associated with the candidate solution.6. A computer program product for use with a stored program computer,the computer program product comprising a computer usable medium havinga computer readable program code embodied therein for generating anoptimized solution for a process in a manufacturing unit, the computerreadable program code performing: a. evaluating at least one solutioncomprising plurality of solution elements associated with the processbased on a set of constraints, the at least one solution representing asequence of the plurality of solution elements, each solution elementhaving one or more attributes, the at least one solution being generatedusing a pre-defined algorithm, wherein the at least one solution elementis evaluated based on the corresponding at least one attribute; b.generating a first violation metric and a first state for each solutionelement, the first state and the first violation metric being generatedcorresponding to at least one constraint based on a first pre-definedcriterion, wherein the first state is generated based on a first stateof a solution element positioned prior to the solution element in thesequence for the at least one constraint; c. generating a candidatesolution by modifying at least one sequence positions of one or moresolution elements in the at least one solution; d. evaluating thecandidate solution at: i. each solution element corresponding to eachchange point, a solution element at each change point being evaluatedfor the at least one constraint based on a first state of a solutionelement positioned prior to the solution element in the sequence, andthe corresponding at least one attribute, wherein a new state and a newviolation metric corresponding to the at least one constraint for thesolution element at the corresponding each change point is generated;ii. one or more solution elements positioned subsequent to each changepoint corresponding to the at least one constraint, wherein theevaluation at the one or more solution elements is performed till atleast one solution element of the one or more solution elements has anew state different from the corresponding first state, wherein eachsolution element is evaluated based on the corresponding at least oneattribute and a violation metric is generated for each of evaluatedsolution element; e. testing the candidate solution based on anacceptance criterion; f. updating the first violation metric and thefirst state associated with each solution element corresponding to eachchange point and the at least one solution element of the one or moresubsequent solution elements when the candidate solution is selectedafter testing; g. repeating steps c, d, e and f for a pre-defined numberof times, wherein the at least one solution is replaced with theselected candidate solution; and h. outputting the optimized solution,wherein the optimized solution is the selected candidate solution afterthe pre-defined number of iterations.
 7. The computer program product ofclaim 6, wherein at least one solution element of the at least onesolution is evaluated when at least one attribute of the at least onesolution element matches a pre-defined attribute.
 8. The computerprogram product of claim 6, wherein at least one solution element of thecandidate solution is evaluated when at least one attribute of the atleast one solution element matches a pre-defined attribute.
 9. Thecomputer program product of claim 6, wherein updating the firstviolation metric is performed based on a first pre-defined criterion.10. The computer program product of claim 6, wherein the acceptancecriterion comprises a cost associated with the candidate solution isless or equal to a pre-defined cost, the calculation of the costcomprises: a. transforming each violation metric associated with eachsolution element of the candidate solution, each violation metric beingtransformed based on a pre-defined function, each violation metric beingassociated with a corresponding constraint; b. normalizing eachtransformed violation metric; c. scaling each normalized violationmetric based on a priority associated with the corresponding constraint;and d. summing the scaled violation metric corresponding to eachconstraint to generate the cost associated with the candidate solution.11. A system comprising: i. a processor configured for: a. evaluating atleast one solution comprising plurality of solution elements associatedwith the process based on a set of constraints, the at least onesolution representing a sequence of the plurality of solution elements,each solution element having one or more attributes, the at least onesolution being generated using a pre-defined algorithm, wherein the atleast one solution element is evaluated based on the corresponding atleast one attribute; b. generating a first violation metric and a firststate for each solution element, the first state and the first violationmetric being generated corresponding to at least one constraint based ona first pre-defined criterion, wherein the first state is generatedbased on a first state of a solution element positioned prior to thesolution element in the sequence for the at least one constraint; c.generating a candidate solution by modifying at least one sequencepositions of one or more solution elements in the at least one solution;d. evaluating the candidate solution at: i. each solution elementcorresponding to each change point, a solution element at each changepoint being evaluated for the at least one constraint based on a firststate of a solution element positioned prior to the solution element inthe sequence, and the corresponding at least one attribute, wherein anew state and a new violation metric corresponding to the at least oneconstraint for the solution element at the corresponding each changepoint is generated; ii. one or more solution elements positionedsubsequent to each change point corresponding to the at least oneconstraint, wherein the evaluation at the one or more solution elementsis performed till at least one solution element of the one or moresolution elements has a new state different from the corresponding firststate, wherein each solution element is evaluated based on thecorresponding at least one attribute and a violation metric is generatedfor each of evaluated solution element; e. testing the candidatesolution based on an acceptance criterion; f. updating the firstviolation metric and the first state associated with each solutionelement corresponding to each change point and the at least one solutionelement of the one or more subsequent solution elements when thecandidate solution is selected after testing; g. repeating steps c, d, eand f for a pre-defined number of times, wherein the at least onesolution is replaced with the selected candidate solution; and h.outputting the optimized solution, wherein the optimized solution is theselected candidate solution after the pre-defined number of iterations.ii. a memory configured for storing the first state, the first violationmetric, the new state, the new violation metric, the at least onesolution, the candidate solution, the acceptance criterion, wherein thememory operatively coupled with the processor.
 12. The system of claim11, wherein at least one solution element of the at least one solutionis evaluated when at least one attribute of the at least one solutionelement matches a pre-defined attribute.
 13. The system of claim 11,wherein at least one solution element of the candidate solution isevaluated when at least one attribute of the at least one solutionelement matches a pre-defined attribute.
 14. The system of claim 11,wherein updating the first violation metric is performed based on afirst pre-defined criterion.
 15. The system of claim 11, wherein theacceptance criterion comprises a cost associated with the candidatesolution is less or equal to a pre-defined cost, the calculation of thecost comprises: a. transforming each violation metric associated witheach solution element of the candidate solution, each violation metricbeing transformed based on a pre-defined function, each violation metricbeing associated with a corresponding constraint; b. normalizing eachtransformed violation metric; c. scaling each normalized violationmetric based on a priority associated with the corresponding constraint;and d. summing the scaled violation metric corresponding to eachconstraint to generate the cost associated with the candidate solution.